Why is the character χ(g) = tr(ρ(g)) constant on conjugacy classes of G?
Think about your answer, then reveal below.
Model answer: For any h ∈ G, χ(hgh⁻¹) = tr(ρ(hgh⁻¹)) = tr(ρ(h)ρ(g)ρ(h)⁻¹) = tr(ρ(g)) = χ(g), using the cyclic property of trace: tr(ABA⁻¹) = tr(A⁻¹AB) = tr(B).
The trace's invariance under conjugation is a linear algebra fact that becomes profoundly useful here. It means characters carry no more information than one value per conjugacy class, which drastically reduces the amount of data needed. For S₃ with 3 conjugacy classes, a character is determined by just 3 numbers rather than 6.
Question 2 True / False
Two non-isomorphic irreducible representations over ℂ can have the same character.
TTrue
FFalse
Answer: False
Over an algebraically closed field of characteristic zero, characters completely determine irreducible representations up to equivalence. If χ_ρ = χ_σ as functions on G, then ρ ≅ σ. This is a consequence of the orthogonality relations: distinct irreducible characters are orthogonal in the class function inner product, so they cannot be equal unless they correspond to the same representation.
Question 3 Multiple Choice
If ρ is a representation of degree n, what is χ_ρ(e), where e is the identity element?
A0
B1
Cn
D|G|
Since ρ(e) = Iₙ (the n×n identity matrix), we have χ_ρ(e) = tr(Iₙ) = n. So the character evaluated at the identity gives the dimension of the representation. This is a useful quick check and means the first column of any character table lists the dimensions of the irreducible representations.
Question 4 Multiple Choice
If V ≅ W₁ ⊕ W₂ as representations, how is χ_V related to χ_{W₁} and χ_{W₂}?
Aχ_V = χ_{W₁} · χ_{W₂}
Bχ_V = χ_{W₁} + χ_{W₂}
Cχ_V = χ_{W₁} − χ_{W₂}
DThere is no general relationship
The trace of a block-diagonal matrix is the sum of the traces of the blocks. If V = W₁ ⊕ W₂ and we choose a basis adapted to this decomposition, then ρ(g) is block-diagonal, and tr(ρ(g)) = tr(ρ₁(g)) + tr(ρ₂(g)). So characters are additive under direct sums: χ_{V⊕W} = χ_V + χ_W. This additivity is what makes characters so useful for decomposition — the multiplicities in V ≅ ⊕ nᵢVᵢ can be extracted from χ_V using inner products.